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An investigation of the Riemann zeta-function by physical methods
Usually, when we mention the Riemann zeta-function, the famous Riemann hypothesis (RH) comes to memory, which says that the real parts of the nontrivial zeros of the zeta-function is 1/2. By the way, mathematicians have not yet been able to find its evidence (or refutation). This result is so important (it is related to the distribution of prime numbers) that Clay's mathematical institute has included RH in the number of the most important problems of the millennium. However, in physical applications, the Riemann zeta-function appears much more often without any mention of RH. As an example, perhaps not the best, we mention the problem of regularization of divergent expressions of field theory-the so-called zeta-regularization of S. Hawking. Mathematicians have long been accustomed to the fact that if we represent the final result of the theory in the form of an expression containing the zeta-function, then one does not have to worry, that it, being written in another form, may contain divergence, i.e. be meaningless. This is due to one surprising feature of the zeta-function - to "absorb" infinity into itself, i.e. to ascribe to the expressions, at first sight, divergent, finite values. For example zeta(0)=1+1+1+1+...=-0.5 zeta(-1)=1+2+3+4+5+...=-1/12 However, this fact that did not cause surprise of the mathematicians surprised the non-specialists 1. An attempt to comprehend the above results was undertaken in 2. The meaning of the last paper is to represent the calculation of the zeta function as the result of the operation of a certain Turing machine (MT) the role of a tape of which plays a numerical axis, and the role of a head plays some physical particle which is moving in accordance with the equations of motion determined by the divergent expressions on the right-hand side of formulas given above. Since partial sums in the expression for the second formula determine the path traveled by the particle at constant acceleration, it is necessary to introduce into the equations of motion the source of this acceleration, or of gravity according to Einstein's equivalence principle. In other words, for the equations of motion of the particle, one should choose the equations consistent with the general theory of relativity of Einstein with a suitable source. After solving them, we define the metric on the numerical axis in which the motion of the particle will no longer cause surprise because the final path that a particle will pass in an infinite time will be finite. The final expression (-1/12) for the path is obtained if we take into account the curvature of the metric of the numerical axis in accordance with the solution of the Einstein equations. In fairness, it should be noted that the result in this paper differs from the exact one by about 3% due to the fact that instead of the relativistic expression , the nonrelativistic expression was used for acceleration of the particle. (In the opposite case, the equations could not be solved) From the point of view of the theory of the Turing machine, the result obtained means the inclusion infinity in the number of admissible values of the counting time. Earlier infinite time meant non-computability of the problem. This is true, since summation of a divergent series on an ordinary Turing machine is related to non-computable problems. Therefore, the MT described in the paper refers to the so-called relativistic MT 3. In development of these ideas the calculation of the zeta function of the complex argument 4 was performed and the RH was proved 5. In addition, the idea was expressed that computation, like motion, can change the geometry of the numerical continuum, and moreover the recognized system of Euclidean postulates should be changed to conform with the above formulas 6. References 1. D. Berman, M. Freiberger, Infinity or -1/12?, + plus magazine, Feb. 18, 2014, http://plus.maths.org/content/infinity-or-just-112. '''2. Y.N. Zayko, The Geometric Interpretation of Some Mathematical Expressions Containing the Riemann ζ-Function, Mathematics Letters,2016; 2(6): 42-46. 3. I. Nemeti, G. David, Relativistic Computers and the Turing Barrier. Applied Mathematics and Computation, 178, 118-142, 2006. 4. Y. N. Zayko, Calculation of the Riemann Zeta-function on a Relativistic Computer, Mathematics and Computer Science,2017; 2 (2): 20-26. 5. Y. N. Zayko, The Proof of the Riemann Hypothesis on a Relativistic Turing Machine, International Journal of Theoretical and Applied Mathematics. 2017; 3(6): 219-224,http://www.sciencepublishinggroup.com/j/ijtam, doi: 10.11648/j.ijtam.20170306.17. 6. Y. N. Zayko, The Second Postulate of Euclid and the Hyperbolic Geometry, International Journal of Scientific and Innovative Mathematical Research (IJSIMR), Volume 6, Issue 4, 2018, PP 16-20. http://dx.doi.org/10.20431/2347-3142.0604003; arXiv: 1706.08378, 1706.08378v1 math.GM)